Microemulsion Polymerization. 2. Influence of Monomer Partitioning, Termination, and Diffusion Limitations on Polymerization Kinetics

Macromolecules 2001, 34, 3233-3244 3233

Renko de Vries, Carlos C. Co, and Eric W. Kaler* Center for Molecular and Engineering Thermodynamics, Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716 Received July 18, 2000; Revised Manuscript Received January 2, 2001

ABSTRACT: We investigate the polymerization kinetics of microemulsions prepared with the cationic surfactant dodecyltrimethylammonium bromide and the hydrophobic monomers n-butyl methacrylate, tert-butyl methacrylate, n-hexyl methacrylate, and styrene. Our previous model for microemulsion polymerization kinetics cannot account for the kinetics of these systems. Using the results of small- angle neutron scattering monomer partitioning studies and an extended kinetic model to analyze the data, the failure of the original kinetic model is shown to be due to a combination of nonlinear monomer partitioning, nonnegligible bimolecular termination, and, in some cases, diffusion limitations to propagation.

1. Introduction agreement with our experimental observations. More- over, full quantitative agreement between theory and In the first paper of this series we investigated the experiment was found for all conversions, using litera- thermodynamics of polymerizing microemulsions. The ture values for the various parameters. monomer concentration in polymer particles formed during microemulsion polymerization was determined Previously we have suggested that in some cases the failure of our simple kinetic model2 may be due to the using small-angle neutron scattering (SANS), and a 7 simple thermodynamic monomer partitioning model pH dependence of the initiation efficiency of unbuffered was developed that explains our experimental findings. persulfate initiators that are typically used. However, for styrene microemulsion polymerizations initiated These results set the stage for the present paper, the with the pH-insensitive cationic initiator 2,2′-azobis(2- aim of which is to develop a better understanding of the amidinopropane) hydrochloride (V50), it was found that kinetics of microemulsion polymerization of hydrophobic the rate maximum still occurs at ∼20%. Similarly, rate monomers initiated with water-soluble initiators. A maxima at conversions significantly below 39% have number of attempts have already been made at model- been found for a number of other systems initiated with ing the microemulsion polymerization kinetics for these 1-6 V50 and buffered persulfate initiators. Hence, while pH systems, but as yet there is no generally applicable effects may be important in some cases, they are not model. Models that have been proposed each address a the only cause of the observed differences. specific system. Instead, it is more likely that the failure of our An elaborate model for the kinetics of styrene micro- original kinetic model for systems other than the base emulsion polymerization has been proposed by Guo et case of C MA/DTAB/DDAB microemulsions is due to 5 6 al. This model accounts for the observed kinetics up to nonnegligible bimolecular termination, nonlinear mono- fairly high conversions but fails to describe the observed mer partitioning, or diffusion-limited propagation due growth of the particle number density. To account for to glass transition. In part 1 of this series, we have the experimentally observed kinetics, the model requires shown that monomer partitioning may be significantly that the rate constant for the capture of an aqueous nonlinear. However, as it turns out, nonlinear monomer phase free radical by a monomer-swollen micelle is partitioning alone is insufficient to explain the failure many orders of magnitude smaller than that for capture of our previous kinetic model for styrene microemulsion by a polymer particle. A very similar model was recently polymerizations. This suggests that termination and/ 6 proposed by Mendizabal et al. or diffusion limitations may also be important. We have previously studied the microemulsion po- Here, we study the polymerization kinetics of n-butyl lymerization of n-hexyl methacrylate (C MA) in micro- 6 methacrylate (nC4MA), tert-butylmethacrylate (tC4MA), emulsions prepared using a mixture of dodecyltrimeth- C6MA, and styrene microemulsions prepared with DTAB. ylammonium bromide (DTAB) and didodecyldimethyl- We have previously studied the monomer partitioning 1-4 ammonium bromide (DDAB) surfactants. A simple behavior of these four systems in part 1 of this series. 2 kinetic model was developed, and solved analytically, The choice of these monomers allows us to indepen- based on the simplifying assumptions that bimolecular dently probe the effects of monomer water solubility, termination is negligible and that the monomer con- polymer glass transition temperature, and propagation centration in the growing particles decreases linearly rate constant on the polymerization kinetics. We also with increasing conversion. With these assumptions, a extend our previous kinetic model by including both the rate maximum is predicted to occur at 39% conversion, observed nonlinear monomer partitioning and account- independent of any characteristic of the system, in good ing for the possibility of bimolecular termination. Using the extended kinetic model to analyze the data, we * Corresponding author. Tel (302) 831-3553; Fax (302) 831-8201; demonstrate that for nC4MA and C6MA microemulsions E-mail [email protected]. the failure of our previous kinetic model is due to

Figure 2. Experimental and model calculated kinetics for C6- MA/DTAB microemulsions initiated with varying concentrations of V50: (O) 0.044 mM and (0) 0.015 mM that correspond Figure 1. Sample withdrawal system used in measuring respectively to 0.045 and 0.015 wt % relative to the amount of polymerization kinetics. In position A, the sample is with- monomer in the microemulsion. drawn directly into 20 mL ampules for gravimetric analysis. In position B, the residue in the sampling lines are flushed. sample as it enters the ampules, and freezing in liquid nitrogen is done only as a precaution. Even when kept at 60 °C for an additional 30 min, no additional polymerization can be de- nonlinear monomer partitioning. For styrene micro- tected in the sealed samples that have been exposed to oxygen. emulsions, we argue that bimolecular termination can- Condensate on the exterior of the ampules is washed off with not be neglected. In addition, diffusion limitations to acetone, and then the amount of sample in the ampules is propagation may be important for both tC4MA and measured by difference. Water and monomer are evaporated styrene microemulsion polymerizations even at low by positioning the ampules horizontally overnight under a conversions. gently blowing air stream. A volatile solvent (e.g., acetone) in We consider chemical initiation by water-soluble which the surfactant is insoluble is then used to extract initiators only. Initiation by oil-soluble initiators is more residual volatiles from the cake, and the ampules are air-dried again after rotating them 180° in the rack. After thorough air- complicated. The radical pair that presumably forms in drying, the samples are dried further in a vacuum oven either the micelles or in the polymer particles may initially kept at 30 °C for 12 h and then ramped up to 60 °C either recombine or the radicals may exit the micelles for 6 h. The samples are cooled to ambient temperature and or particles in a way that is expected to depend then reweighed. The weighings are performed carefully on a sensitively on the micelle and particle properties. This four-digit balance, and the precision of the 30-40 conversion complication is absent for water-soluble initiators. measurements obtained for each reaction is routinely better than (0.5%. The accuracy of these measurements is within 2. Materials and Methods (1% and is limited by the accuracy in preparing the original microemulsion. All monomers (nC4MA, tC4MA, C6MA, and styrene) were To extract rates of polymerization, the conversion vs time purchased from Scientific Polymer and vacuum distilled to data are differentiated using Craven and Wahba’s cross- remove inhibitors. The distilled monomers were stored in a validation smoothing spline algorithm.8 Statistical validity is freezer for less than 5 days before use. DTAB (99+%) from then examined using a bootstrapping algorithm9 to account TCI and V50 initiator (98.8%) provided by Wako were used for experimental errors in measuring both conversion and as received. Prior to preparing the microemulsions, deionized time. For each iteration of this bootstrapping procedure, all water (18.3 MΩ cm) was boiled under vacuum for at least 30 data points are resampled from normal distributions centered min, and the monomers were sparged with ultrahigh-purity around the original data points with standard deviations of N2 (<1 ppm of O2) for 15 min to remove oxygen and then 1 s and 0.5% respectively for the time and conversion. The sealed. Thirty grams of DTAB was weighed into a 500 mL resampled data set is then differentiated using the cross- water jacketed reactor equipped with a stirrer, condenser, and validation smoothing spline algorithm after which the rates heated reactor head. The reactor was sealed and then purged at a few randomly chosen conversions are recorded together thoroughly with four cycles of vacuum and ultrahigh-purity with the position of the rate maximum. One thousand itera- N2. The deoxygenated water and monomers (212.4 and 6.6 g, tions of this bootstrapping procedure are performed for each respectively) were then injected into the reactor. This composi- data set to obtain error estimates for the position of the tion corresponds to a surfactant weight fraction (γ)of12wt% maximum rate of polymerization. The large set of rates and a monomer weight fraction on a surfactant-free basis (R) recorded at random conversions are then plotted, e.g., Figure of 3 wt %. After heating to 60.0 °C, polymerization of the 3, to permit a qualitative assessment of the agreement between microemulsion was initiated by injecting1gofaV50solution theory and experiment. prepared using deoxygenated water. After an induction period The aqueous solubilities of the monomers at 60 °C were of less than 30 s, the microemulsions became faintly bluish, measured using UV absorbance. Samples containing deionized indicating the onset of polymerization. water and a slight excess of vacuum distilled monomer were Over the course of the reaction, ∼5 mL samples are taken equilibriated and then allowed to phase separate over4hat out from the reactor directly into preweighed 20 mL ampules 60.0 °C. Approximately 10 g of the aqueous phases was taken using the sampling valve shown in Figure 1. The ampules are directly into syringes containing ∼10 g of ethanol. The immediately sealed with a septa and then frozen in liquid monomer concentration was then measured by monitoring the nitrogen. Exposure to oxygen instantaneously quenches the UV absorption at 205 and 220 nm for nC4MA, tC4MA, and C6- Macromolecules, Vol. 34, No. 10, 2001 Microemulsion Polymerization. 2 3235

where [I] is the initiator concentration and kd is the first- order rate constant for initiator decomposition. The reaction of a primary free radical (I•) with a first monomer (M) is reported to occur very rapidly, essentially in the diffusion-limited regime.14 The resulting species (IM•) either may be amphiphilic enough to directly enter a monomer-swollen micelle and start propagating or require further propagation steps in the aqueous phase. This is mechanistically similar to the Maxwell-Morisson model15 for radical entry in emulsion polymerizations. If aqueous phase propagation is necessary, the free radical concentration may build up to levels where aqueous phase termination can no longer be neglected. Chain growth may stop by either bimolecular termination or chain transfer to monomer. In view of the high concentration of surfactants, particle coalescence is expected to be negligible, and bimolecular termination by this mechanism can be excluded. Because of the Figure 3. Experimental and model rate vs conversion profiles small particle sizes, microemulsion polymerization is obtained by differentiating the data shown in Figure 2. expected to obey zero-one kinetics. Therefore, the main mode of bimolecular termination for growing chains in Table 1. Monomer Physical Properties microemulsion polymerization is the entry of a second C6MA nC4MA tC4MA styrene free radical into a growing particle, followed by instan- • water solubility at ∼0.4a 3.4a 4.3a 4.6b taneous termination. Monomeric radicals (M ) generated 60 °C (mM) by chain transfer to monomer may either continue 20 20 Tg (°C) -5 20 128 106 propagating in the same particle or exit to the aqueous -1 -1 28 28 33 34,35 kp at 60 °C (M s ) 995 1015 1140 342 phase. For the small particles produced by microemul- a Measured as described in section 2. b Interpolated from data sion polymerization, exit is by far the most probable fate • by Lane.10 of the transfer-generated (M ) molecules. The fundamental equation for the rate of the micro- MA and at 203 and 248 nm for styrene. Linear calibration curves were obtained using standard solutions with the same emulsion polymerization reaction is water/ethanol ratio as the unknown. The styrene aqueous solubility of 4.6 ( 0.3 mM shown in Table 1 is in agreement (part) ∂f kpCmon N* with the value of 5.1 mM obtained by interpolating measure- ) (2) ments done by Lane.10 Likewise, our measurement of 3.4 ( ∂t M0 0.3 mM for the aqueous solubility of nC4MA at 60 °C is consistent with the value of 2.5 mM at 50 °C as measured by This assumes a negligible contribution from aqueous 11 Gilbert et al. The aqueous solubility of C6MA is low and phase propagation. In eq 2, f is the fractional conversion, hampers the reproducibility of our measurements. We estimate M0 is the initial concentration of monomer in moles per an aqueous solubility of ∼0.4 mM for C6MA that is also 12 liter of microemulsion, kp is the propagation rate consistent with group contribution estimates. (part) The glass transition temperatures of microemulsion poly- constant, Cmon is the monomer concentration at the styrene and poly-tC4MA were measured using a Perkin-Elmer locus of polymerization in the growing polymer particles, DSC7, at heating rates of 20 and 10 °C/min. Latexes of and N* is the concentration of propagating radicals. polystyrene and poly-tC4MA were precipitated and washed Note that, in view of our assumption of zero-one repeatedly with an excess of methanol and vacuum-dried at kinetics, N* also equals the concentration of growing 40 °C. Just as Quian et al.13 have observed, an exotherm was particles. Except for M0, all of the quantities on the recorded at the approximate glass transition temperature right-hand side of eq 2 may vary as the reaction during the initial scan of both polymers. Subsequent scans did not exhibit this release of energy that is related to the proceeds. The propagation rate constant kp varies kinetically controlled conformation of the polymers as dis- significantly only if the polymer passes through a glass cussed by Quian et al.13 The glass transition reported in Table transition during the reaction, in which case the reac- 1 are those measured for a heating rate of 10 °C/min, which tion becomes limited by diffusion rather than reaction. are ∼1.5 °C less than that measured at a heating rate of 20 In the absence of a glass transition, kp is approximately °C/min. constant. The concentration of monomer in the growing par- (part) 3. Theory ticles, Cmon , is determined by how the unreacted monomer partitions between the polymer particles and This section briefly reviews our previous kinetic model the large number of coexisting surfactant micelles. We that is now extended to account for nonlinear monomer have previously assumed that for C6MA/DTAB/DDAB partitioning and termination. We also briefly discuss microemulsion polymerization monomer partitioning is possible diffusion limitations to propagation for tC4MA linear,2-4 and styrene microemulsion polymerizations. The basic sequence of events has been outlined in our (part) ) (part) - previous papers.2,3 After adding a water-soluble initiator Cmon Cmon,0(1 f) (3) to the microemulsion, primary free radicals are generated in the aqueous phase at a rate F0, Next, consider the concentration of growing particles, N*. For microemulsion polymerization the ratio of N* F ) 0 2kd[I] (1) to the total number density of dead polymer particles 3236 de Vries et al. Macromolecules, Vol. 34, No. 10, 2001 and monomer-swollen micelles N is4,16 Table 2. Conversion at the Maximum Rate of Polymerization (fmax) for Varying Values of the Kinetic N* - - Model Exponent (b)inEq10 ≈ 10 4-10 3 (4) N bfmax (%) 139 where, to a very good approximation, N equals the 1.2 35 concentration of monomer-swollen micelles. Therefore, 1.4 32 we have previously argued that the entry of an aqueous 1.6 29 phase free radical into a growing particle is a highly improbable event. The capture of aqueous phase, initia- Only for a styrene microemulsion with low monomer tor-derived free radicals by monomer-swollen micelles content (R)3 wt %) far away from the phase boundary can be assumed to be a process that is fast on the time (R)8.2 wt %) did we find linear monomer partitioning scale of the polymerization reaction, even if capture (b ≈ 1). For other microemulsions with compositions requires some aqueous phase propagation. The explicit closer to the phase boundary, monomer partitioning was rate equation for the concentration of aqueous phase, significantly nonlinear (b ≈ 1.4). initiator-derived free radicals can therefore be elimi- If bimolecular termination can still be neglected, the nated using the steady-state approximation. Neglecting equation for the rate of polymerization, now accounting termination in the particles, the rate equation for N* for nonlinear monomer partitioning, can be written as can then be written as ∂f ∂N* ) A(1 - f)bt (10) )F (5) ∂t ∂t with A again given by eq 7. The conversion at the where F)γeffF0, and γeff is an efficiency factor, assumed to be independent of conversion, that accounts for maximum rate is now a function of the exponent b, and possible aqueous phase termination. Note that, in the increasing b from 1 to 1.6 lowers the location of the absence of termination, chain transfer to monomer does maximum from 39% to 29% (Table 2). Hence, nonlinear not affect the total number of growing particles and monomer partitioning can indeed account for a shift of hence does not appear in eq 5. Combining eqs 2 and 5 the location of the rate maximum to lower conversions. ) we find However, for b 1.4, as found in our experiments, the maximum only shifts down to 32%, which is still much ∂f larger than the typical value of ∼20% observed for ) A(1 - f)t (6) ∂t styrene microemulsion polymerization. Therefore, we proceed to also consider the possibility of bimolecular k C(part) F termination. A ) p mon,0 (7) 3.2. Termination in the Particles. Previously, we M0 have argued that termination should be negligible for microemulsion polymerization2 in view of the large the solution of which is excess of micelles over growing polymer particles, as expressed by eq 4. Such an argument would indeed hold 1 f(t) ) 1 - exp - At2 (8) if the small aqueous phase free radicals, including both ( 2 ) the initiator-derived and the transfer-generated radicals, always start propagating upon entering or adsorb- This has a rate maximum at t ) A-1/2, at a conversion ing on a monomer-swollen micelle. However, this surely of f ) 1 - e-1/2 ≈ 0.39.2 For microemulsion polymeriza- depends on the reactivity of these radicals. Free radicals tions of C MA in mixed DTAB/DDAB microemulsions, 6 that are still small enough to have some aqueous phase we have demonstrated that eq 8 quantitatively accounts solubility could enter and exit a large number of swollen for the polymerization kinetics for reasonable values of micelles before eventually propagating in one. This the various parameters, assuming negligible aqueous would greatly enhance the probability for them to phase termination, γ ) 1. Rate maxima at ∼40% eff encounter and terminate a growing polymer particle. conversion have also been reported by Capek and Following the discussion of Maxwell et al.15 for emulsion Juranicova17 for the microemulsion polymerizations of polymerization, this is easily understood considering the ethylhexyl methacrylate. Reanalysis of the kinetic data time scales involved. The typical time scale for propaga- by Full et al.18 for tetrahydrofurfuryl methacrylate tion in a monomer-swollen micelle is microemulsions also reveals a rate maximum at ∼40% conversion. Note that in both cases the monomers have ) (mic) -1 very low water solubilities and low polymer glass τprop (kpCmon ) (11) transition temperatures. However, all other systems that have been studied so far seem to have rate maxima ≈ 3 -1 -1 (mic) ∼ ≈ -3 so for kp 10 M s and Cmon 1M,τprop 10 s. at conversions significantly below 40%, for reasons that There are two types of aqueous phase small free will now be explored. • radicals: initiator-derived radicals (IMn ) and mono- 3.1. Nonlinear Monomer Partitioning. To test the meric free radicals (M•) generated by chain transfer to assumption of linear monomer partitioning, in part 1 monomer. For water-soluble initiators, the initiator- of this series, we have studied monomer partitioning derived small free radicals are surfactant-like molecules. in polymerizing microemulsions. These monomer par- Typical residence times of surfactants in micelles (τres titioning results are correlated by the empirical inter- ≈ 10-6-10-5 s) depend strongly on their aqueous polation formula monomeric solubility. Similar residence times may be expected for small initiator-derived free radicals. An (part) ) (part) - b Cmon Cmon,0(1 f) (9) estimate of the residence time for transfer-generated Macromolecules, Vol. 34, No. 10, 2001 Microemulsion Polymerization. 2 3237 monomeric radicals (M•) in monomer-swollen micelles A more detailed derivation of the rate equation (14) is15 starts from the coupled set of rate equations for the concentrations of aqueous phase free radicals, growing 2 particles, and swollen micelles with adsorbed oligomeric Rmic τ ≈ q (12) free radicals, explicitly taking into account the adsorp- res (aq) - 3Dmon tion desorption process. In such a scheme it is easy to also include a possible difference in the adsorption rate constants of the oligomeric free radicals on particles and ≈ (aq) ≈ -9 where Rmic 3 nm is the micelle radius, Dmon 10 swollen micelles. As shown in Appendix A, this gives m2 s-1 is the diffusion constant of the monomer in the ≈ 3- 4 aqueous phase, and q 10 10 is the partition (part) coefficient of the monomer between the micelles and the pterm kads 1 N* - ) (16) aqueous phase. For these values, τ ≈ 3 × 10 6-3 × (mic) (mic) res pprop k τ k C N 10-5 s. ads res p mon Therefore, an aqueous phase free radical will typically adsorb on and desorb from 102-103 monomer-swollen For diffusion-controlled adsorption, micelles before propagating. Recalling that the ratio of (part) the number of monomer-swollen micelles to growing kads Rpart 3- 4 ) (17) particles is 10 10 , it becomes clear that there is (mic) R indeed a nonnegligible probability that small free kads mic radicals in the aqueous phase might terminate growing particles during microemulsion polymerization. where Rpart is the particle radius. For some small To develop a rate equation for the number of growing radicals, such as the initiator-derived radical species • particles that includes the possibility of termination via (IM ), the residence time in the growing particles may the above mechanism, let pprop be defined as the prob- in fact be so small that the assumption of instantaneous ability that, after a single adsorption-desorption step, termination is invalid. Hence, a further refinement of the oligomeric free radical has propagated in a monomer- eq 16 could be necessary and would have to include an swollen micelle. Likewise, pterm is the probability that, efficiency factor for the termination reaction. after a single adsorption-desorption step, the oligomeric Next we discuss the variation of pterm/pprop as the (part) (mic) free radical has terminated a growing particle. In view reaction progresses. First consider the ratio kads /kads of the discussion above, pprop and pterm are both much ≈ Rpart/Rmic. During the reaction, the monomer swelling less than unity. After infinitely many adsorption- of both particles and micelles decreases. This implies desorption steps, the probability that the oligomeric free that, to a first approximation, the ratio Rpart/Rmic will radical will have propagated is Pprop ) pprop/(pprop + be constant as the reaction proceeds. Therefore, in what (part) (part) pterm). The probability that the oligomeric will have follows, the ratio k /k is assumed to be indepen- ) - ads ads terminated a growing particle is Pterm 1 Pprop. Thus, dent of conversion. While we do have an estimate for a more complete rate equation for the number of the residence time of the transfer-generated radicals, growing particles that accounts for termination is eq 12, there is no analogous estimate for the residence time of the initiator-derived free radicals that are ∂N* surfactant-like. Specifically, there is no model for how )F(P - P ) - 2k C(part)N*P (13) ∂t prop term tr mon term the surfactant residence time depends on the radius of the microemulsion droplets. Nonetheless, the residence - 1 pterm/pprop (part) pterm/pprop time is expected to scale with the water solubility of the )F - 2k C N* (14) • 1 + p /p tr mon 1 + p /p initiator-derived radical (IM ), which in turn depends term prop term prop on the aqueous solubility of the monomer. Moreover, due to their higher rotational entropy, small free radicals In more physical terms, these equations describe the are expected to have a higher reactivity compared to following: Initiator-derived small free radicals are their polymeric counterparts. Effective k values ap- F p generated at a rate . A fraction Pprop of these eventually proximately 4 times higher than the long-chain limit propagate in a monomer-swollen micelle, each increas- are expected on the basis of theoretical calculations.19 ing the number of growing particles by one. The Therefore, in what follows, we do not use a specific remaining fraction P terminate growing polymer term model for τres but instead simply assume that particles, each decreasing the number of growing particles by one. Monomeric radicals (M•) are generated by pterm 1 1 N* chain transfer to monomer at a rate k C(part)N*, where ) (18) tr mon (mic) pprop τterm N ktr is the bimolecular rate constant for chain transfer kpCmon to monomer. A fraction (Pprop) of these continue propagating and do not change N*. The other fraction where τterm ≈ τresRmic/Rpart is a conversion-independent terminates growing polymer particles, with each radical time scale into which the correction factor for kp is decreasing the number of growing particles by two. invariably lumped. A final problem is that the relative Clearly, the crucial parameter is the ratio pterm/pprop.If contributions of the initiator-derived and the transfer- pterm/pprop is zero, eq 13 reduces to eq 5, which neglects generated small free radicals to the termination process termination of growing chains. A crude estimate for are unknown. Part of the problem is that, as mentioned, pterm/pprop is simply there is no detailed model for the residence time of the • initiator-derived species (IMn ). Therefore, in eq 14 we pterm τprop N* 1 N* simply use a single value of pterm/pprop or, equivalently, ≈ ) (15) a single value of τ that characterizes the termination p τ N (mic) N res prop res τreskpCmon process. This parameter should be interpreted as a 3238 de Vries et al. Macromolecules, Vol. 34, No. 10, 2001 characteristic residence time of the “typical” aqueous Unfortunately, the decrease of kp as a function of phase free radical that causes the termination. polymer content has not been measured for tC4MA. According to eq 18, the ratio pterm/pprop that enters the Nevertheless, by analogy with electron spin resonance kinetic equations depends on the concentration of (ESR) data for poly(methyl methacrylate) (Tg ∼105 °C), 24 monomer in the micelles, as well as on the number as reviewed by Yamada et al., we expect that kp should density of micelles, both of which depend on the conver- remain constant up to the volume fraction of polymer sion. Given the monomer concentration in the polymer (∼70%) where the polymer particles turn glassy. particles, for which we assume the functional form of From our monomer partitioning studies, the volume eq 9, these quantities can be calculated using simple fraction of monomer in the polymer particles is ∼30 mass-balance equations. The explicit expressions for the vol % for both styrene and tC4MA at their typical rate micelle number density (N) and the concentration of maxima at ∼20% conversion. Given these results and (mic) monomer within them (Cmon ) are given in Appendix B. the available literature data, it is unfortunately difficult Finally, within the context of the present model, we to estimate the importance of diffusion limitations due can interpret the assumption of Guo et al.,5 that the rate to glass transition effects on the location of the rate constant for radical capture by the micelles, ka′ (using maxima. In the absence of accurate data on the depen- their notation), is many orders of magnitude smaller dence of kp on the polymer weight fraction for styrene than that for capture by the particles, ka. They at- or tC4MA, we refrain from including this effect in our tributed this to the fact that the micelles are highly theory. Instead, we simply demonstrate the qualitative charged. However, in the presence of so much surfactant differences in the kinetic profiles for nC4MA and tC4- the polymer particles are most likely fully covered with MA which, recall, have essentially identical physical surfactant and hence are just as highly charged as the properties with the exception of their glass transition micelles. Even if this were not the case, electrostatics temperatures. From these observations, we then infer alone could not cause such a large difference in the that diffusion limitations may combine with termination capture rates of the small aqueous phase free radicals. effects in shifting the maximum rate to ∼20% conver- This confusion arises because the process of radical sion for styrene. capture by particles and micelles was not clearly That diffusion limitations are the general cause for defined. Radical capture by polymer particles affects the the decline in the propagation rate beyond the glass kinetics through bimolecular termination. This is a fast transition has been argued to be highly improbable by process; i.e., once a small free radical enters a growing Faldi et al.25 Their forced Rayleigh scattering and field- particle, termination follows essentially immediately. gradient NMR measurements of MMA diffusion coef- On the other hand, radical capture by micelles repre- ficients in PMMA are 4 orders of magnitude greater sents the entire sequence of events up to the first than that necessary to achieve agreement with kp values propagation step in a monomer-swollen micelle, which measured using ESR. Gilbert26 compares these and is a slow process. Within the context of our theory, the other measurements of the diffusion coefficient with the ′ ratio ka/ka , of the capture rate constants is simply value expected from ESR measurements of kp in an emulsion polymerization as a function of polymer vol- (part) ume fraction in the particles. There is indeed a large ka kads τres Rpart ) ≈ k C(mic)τ (19) difference between the diffusion coefficients calculated k ′ (mic) τ R p mon res a kads prop mic from kp via the Smoluchowski equation and the direct diffusion coefficient measurements. However, as the which is indeed a large number, as found by Guo et al.5 diffusion coefficient measurements were performed on bulk mixtures of monomer and polymer while the 3.3. Diffusion Limitations to Propagation. For diffusion coefficients calculated from ESR measure- polystyrene and poly(tC MA) that have glass transition 4 ments of k were performed in an emulsion polymeri- temperatures of ∼100 and ∼115 °C, respectively,20 p zation, the accuracy of such comparisons may be com- diffusion limitations to propagation may arise within promised by systematic errors in the measurement of the polymer particles as the polymerization proceeds at polymer content. Gilbert’s graphical comparison shows 60 °C. Hence, we must address the question of whether clearly that this discrepancy in the diffusion coefficients this may also contribute to the shift in the location of can be caused by small errors (∼5%) in the measure- the rate maxima to lower conversions. Note that the ment of the polymer volume fraction. As Faldi et al.25 time scales for diffusion across the aqueous phase and offer no alternative explanation for the generally ob- the surfactant layers of the micelles and polymer served decline in the propagation rate past the glass particles are many orders of magnitude faster than the transition temperature, this apparent discrepancy is time scale of propagation and need no further consid- assumed to be due to experimental errors and will not eration.2 be considered further. For bulk radical polymerizations of styrene, Yamada 21,22 et al. have performed ESR measurements that led 4. Results and Discussion them to conclude that kp is constant up to high conversions at 70 °C. However, qualitative reanalysis of their 4.1. C6MA: Effect of Nonlinear Monomer Parti- raw data indicate that a sharp drop in kp is expected tioning on Kinetics. We have previously studied the 21 beyond 80% conversion. Furthermore, their average microemulsion polymerization kinetics of C6MA with 22 molecular weights are very low (Mn ) 13 000 Da, Mw/ mixed DTAB/DDAB surfactants and found excellent 2 Mn ) 2.32 at 79% conversion) compared to the molecular agreement with our original kinetic model. Here we weights (∼107 Da) of polymers typically obtained from simplify our model system by using only DTAB as the microemulsion polymerizations (part 3). Indeed, the surfactant. To relate these new results to our previous glass transition temperature of polystyrene with Mn ) kinetic studies, we first compare the microemulsion 23 13 000 is only ∼90 °C compared to Tg ∼ 100 °C for polymerization kinetics of the mixed microemulsions typical polystyrenes with Mn > 50 000. with that of microemulsions prepared with DTAB only. Macromolecules, Vol. 34, No. 10, 2001 Microemulsion Polymerization. 2 3239

This also allows us to study the influence that the MA. For the dissociation rate constant of the V50 surfactant has on the kinetics of microemulsion polym- initiator at 60 °C we use the value quoted by the -5 -1 27 erization. manufacturer, kd ) 3 × 10 s . As for C6MA/DTAB/ Figure 2 shows the conversion as a function of time DDAB microemulsion polymerization, aqueous phase for C6MA/DTAB microemulsions initiated with V50 at termination is neglected, and an initiator efficiency of 60 °C for two initiator concentrations. The conversion γeff ) 100% is assumed. The monomer concentration data are analyzed as described in section 2 to give the initially present in the polymer particles was measured (part) ) rate of polymerization as a function of conversion as to be Cmon,0 2.96 M in part 1. Finally, the macro- shown in Figure 3. scopic monomer concentration in the microemulsion was ) Remarkably, a mere change of the surfactant mixture M0 0.18 M. For the theoretical curves shown in ) -1 -1 causes a shift in the location of the rate maximum from Figures 2 and 3, a single value of kp 940 M s was ∼40% for the mixed DTAB/DDAB microemulsions down used for the propagation rate constant. This value is ) -1 -1 to about ∼30% for the DTAB-only microemulsions. As perfectly consistent with the value of kp 995 M s discussed previously, the extent of radical termination that we have previously obtained by interpolating is affected mainly by the physical properties of the accurate pulsed-laser polymerization data for other monomer and initiator rather than by the properties of linear alkyl methacrylates,28 remembering that, for the microemulsion droplets. Hence, it is highly unlikely example, the value of the initiator decomposition rate that a change of the surfactant mixture would invalidate constant is subject to considerable uncertainty. the assumption of negligible termination that worked The conversions at the maximum rate of polymeri- so well for the case of the mixed DTAB/DDAB micro- zation are f ) 30 ( 3% and f ) 34 ( 2% respectively for emulsions. More likely, the change in surfactant affects the lowest and the highest value of the initiator con- the microemulsion thermodynamics, i.e., monomer par- centration. To within experimental error, these mea- titioning. surements are in agreement with the theoretical value Recall that in the original kinetic model for the mixed of 32% for b ) 1.4 (see Table 2). microemulsions linear monomer partitioning was as- At higher conversions, the theoretical curves overes- sumed. However, the measurements in part 1 show that timate the rate of polymerization, especially for the the monomer partitioning for polymerizing C6MA/DTAB lowest initiator concentration. This is similar to earlier microemulsions is significantly nonlinear (b ≈ 1.4). measurements of the mixed surfactant microemulsions Therefore, we hypothesize that the observed differences when compared to the previous kinetic model. Presum- are caused by a difference in monomer partitioning. The ably, the deviations at higher conversions may be due mixed DTAB/DDAB system presumably has linear to a small amount of termination, either in the aqueous monomer partitioning, whereas the DTAB-only system phase or in the particles. As we have found for the case has significantly nonlinear monomer partitioning, as of C6MA/DTAB/DDAB microemulsion polymerizations, measured experimentally. Unfortunately, it has not in comparing the kinetic model to the experimental been possible to confirm that the monomer partitioning data, there is no way to accommodate low initiator for the mixed DTAB/DDAB microemulsions is indeed efficiencies. In the present case, if we were to assume linear. The reproducibility of SANS monomer partition- that initiation efficiencies are low due to some initial ing studies for these microemulsions has so far been aqueous phase propagation, as suggested by the Max- hampered by the tendency of C6MA latex particles to well-Morisson model15 for entry in emulsion polymer- cream when prepared from DTAB/DDAB microemul- ization, our fitted values for kp would have been sions in D2O. unreasonably high. Despite this experimental difficulty, the expected 4.2. Styrene: Effect of Termination and/or Dif- difference in the monomer partitioning behavior of the fusion Limitations on Kinetics. Next consider the mixed system as compared to the DTAB-only system polymerization kinetics of styrene. Figure 4 shows the can be explained using the thermodynamic model conversion as a function of time for styrene/DTAB developed in part 1. This model, together with experi- microemulsions containing R)3 wt % styrene initiated mental monomer partitioning results for styrene, indi- with V50 at 60 °C for three initiator concentrations. cates that monomer partitioning becomes increasingly Again, the conversion data are analyzed as described nonlinear close to the phase boundary. Indeed, on a in section 2 to give the rate of polymerization as a surfactant-free basis, the DTAB-only microemulsion function of conversion, shown in Figure 5. consists of R)3wt%C6MA, which is very close to the Consistent with data for styrene microemulsion po- phase boundary that lies at R)3.2 wt %. Upon lymerization obtained by others,5,16,29-32 we find rate replacing 30% of the DTAB surfactant with DDAB to maxima at ∼20% conversion. In part 1 of this series, obtain the mixed system, the phase boundary shifts to we have measured highly linear monomer partitioning R)7.5 wt %. Hence, it is likely there is a more linear profiles for styrene/DTAB microemulsions containing monomer partitioning profile for our previous kinetic R)3 wt % styrene. Therefore, the failure of the simple studies that were based on C6MA/DTAB/DDAB micro- theory to account for the kinetics is presumably due to emulsions containing R)5 wt % monomer. nonnegligible biradical termination and/or diffusion To demonstrate that nonlinear monomer partitioning limitations to propagation. is the cause of the observed differences, we compare our While we have shown that significant aqueous phase data with the modified kinetic model (eq 10) that still termination is inconsistent with the observed kinetics neglects biradical termination but accounts for non- of C6MA, this is not necessarily true for styrene. The linear monomer partitioning. Theoretical curves for the aqueous phase solubility of styrene (Table 1) is ap- kinetics are obtained by numerical integration of eq 10, proximately 10 times higher than that of C6MA while using b ) 1.4 as found experimentally. All parameters its propagation rate constant is only one-third that of for the kinetic theory have been measured indepen- C6MA. These factors result in a much higher concentra- dently, except for the propagation rate constant of C6- tion of small styrene-based free radicals in the aqueous 3240 de Vries et al. Macromolecules, Vol. 34, No. 10, 2001

Figure 4. Experimental and model calculated kinetics for styrene/DTAB microemulsions containing R)3 wt % mono- Figure 6. Model predicted variation of the conversion at the mer on a surfactant-free basis. The microemulsions were maximum rate of polymerization with respect to the charac- initiated with (O) 0.49 mM, (4) 0.24 mM, and (0) 0.061 mM teristic time for termination (τ ) for styrene/DTAB micro- of V50 that correspond respectively to 0.5, 0.25, and 0.0625 term emulsions containing R)3 wt % monomer. Curves A and B wt % of the monomer in the microemulsion. - - correspond to initiator radical fluxes of F)5 × 10 9 Ms 1 and F)5 × 10-8 Ms-1, respectively.

of 10 shifts the predicted position of the maximum by only 1%. The positions of the rate maxima based on the experimental data in Figure 5 are at 16 ( 4%, 24 ( 5%, and 23 ( 2%, respectively, for decreasing initiator concentrations. From the average value of 21 ( 3%, the characteristic time scale for termination is estimated -6 to be τterm ) (1.9 ( 0.6) × 10 s; that is consistent with the estimates described in section 3. The initiation efficiencies (γeff) are then estimated by fitting the experimental rate vs conversion data to the kinetic model. The solid curves in Figures 4 and 5 are the result -6 of model calculations using τterm ) 1.9 × 10 s and initiation efficiencies of 80%, 60%, and 55% for increasing initiator concentrations. At conversions beyond ∼50%, the theoretical model overpredicts the polymerization rate. However, as the monomer partitioning measurements demonstrate (part 1), there is no doubt that diffusion limitations due to glass transition are Figure 5. Experimental and model rate vs conversion profiles important at such higher conversions. obtained by differentiating the data shown in Figure 4. The slight decrease of the initiation efficiencies with increasing initiator concentrations does indeed point to phase. Therefore, in comparing the data to our kinetic some aqueous phase termination. However, these high model, an initiation efficiency parameter (γeff) that initiation efficiencies (55-80%) are clearly inconsistent accounts for possible aqueous phase termination is used. with the aqueous phase chemistry suggested by the The only other unknown parameter is the characteristic Maxwell-Morisson15 model for radical entry in emul- time scale for termination (τterm ≈ τresRmic/Rpart), which sion polymerizations. Assuming a critical degree of is independent of the initiator concentration. The rate polymerization for entry of z ) 2, the Maxwell- constant for chain transfer to monomer is not known to Morisson theory would predict extensive aqueous phase the same degree of accuracy as the propagation rate termination for styrene, leading to an initiation ef- -1 constant but is estimated to be about ktr ) 0.02 M ficiency of only a few percent. - s 1.20 For the present styrene microemulsion composi- The microemulsions of interest here are concentrated tion, monomer partitioning is linear (b ≈ 1) and the dispersions of monomer-swollen micelles with intermi- initial monomer concentration in the polymer particles cellar distances of ∼10 nm.4 Therefore, small free (part) ) is Cmon,0 1.67 M (part 1). Finally, the macroscopic radicals in the aqueous phase are never further than a concentration of monomer in the microemulsion is M0 few nanometers from a monomer-swollen micelle. More- ) 0.18 M. over, for typical initiator concentrations of ∼0.1 mM, As shown in Figure 6, the position of the rate the number of free radicals after characteristic reaction maximum in our kinetic model depends mainly on the times of ∼1000 s is only ∼10-6 molecules/nm3, 1000 value of the characteristic time scale for termination times less than the number of micelles. Thus, aqueous τterm and is almost independent of the initiator concen- free radicals in microemulsions encounter monomer tration. Increasing the initiator concentration by a factor much more frequently than they would when confined Macromolecules, Vol. 34, No. 10, 2001 Microemulsion Polymerization. 2 3241

Figure 7. Experimental and model calculated kinetics for Figure 8. Experimental and model rate vs conversion profiles styrene/DTAB microemulsions containing R)7 wt % mono- obtained by differentiating the data shown in Figure 7. mer on a surfactant-free basis. The microemulsion was initiated with (O) 0.57 mM of V50 that corresponds to 0.25 wt % concentrations (b ) 1.4) cannot lower the location of the of the amount of monomer in the microemulsion. maximum rate to below 30% conversion. From this result, we conclude that termination effects are impor- to the aqueous phase as assumed in the Maxwell- tant for styrene/DTAB microemulsion polymerizations. Morisson theory. Hence, we conclude that even for Clearly, this description of the termination process styrene microemulsion polymerization, radical capture is still highly approximate in the sense that the char- by the monomer-swollen micelles is fast and aqueous acteristic time scale for termination (τterm) has been phase termination is limited. Significantly, biradical assumed to remain constant throughout the reaction termination can fully account for the position of the and is equal for all species (M• and IM•) involved. maximum rate of polymerization (∼20%) for styrene Aqueous phase termination has also been assumed to microemulsions. be independent of conversion even though the overall Up to this point, our analysis has ignored the pos- radical concentration increases with conversion. Fur- sibility that the biradical termination effects we have thermore, the possibility remains that termination just described may be simultaneously accompanied by effects may be accompanied by diffusion limitations to diffusion limitations in determining the location of the propagation for styrene/DTAB microemulsions at lower rate maximum. To explore this possibility, we take monomer loadings, e.g., the previous experiments per- advantage of the monomer partitioning measurements formed at R)3 wt %. Nonetheless, this analysis is as (part 1) for styrene/DTAB/D2O microemulsions at higher complete as is warranted by the current experimental monomer loadings. These measurements show that, for situation. R) styrene/DTAB/D2O microemulsions at 7.5 wt %, the 4.3. nC4MA and tC4MA: Effect of Diffusion Limi- polymer particles contain at least 40 vol % of monomer tations Due to Glass Transition. DTAB-based mi- up to 30% conversion. Thus, diffusion limitations cannot croemulsions of nC4MA and tC4MA have physical play a role in determining the location of the maximum properties that make them ideal model systems for polymerization rate for styrene/DTAB microemulsions probing the effects of diffusion limitations to propaga- close to the phase boundary, although they surely play tion on the kinetics of microemulsion polymerizations. a role at higher conversions. In part 1, we have shown that the monomer partitioning Figures 7 and 8 show the kinetic profiles for a styrene/ profiles for these butyl methacrylates are essentially R) DTAB/H2O microemulsion with 7 wt %. The rate identical. Moreover, as shown in Table 1, nC4MA and maximum, which still occurs at a low conversion of 21 tC4MA have essentially identical water solubilities and ( 4%, corresponds to a characteristic time scale for propagation rate constants but polymer glass transition -6 termination of τterm ) (3-9) × 10 s. An intermediate temperatures20 that are respectively ∼40 °C below and -6 value of τterm ) 5 × 10 s was used in conjunction with ∼60 °C above the reaction temperature of 60 °C. This (part) ) an initiation efficiency of 75% and Cmon,0 3.08 M to enables us to isolate the effects of glass transition and calculate the solid curves shown in Figures 7 and 8. further explore the possibility that diffusion limitations Observe that at conversion beyond ∼50% the theoretical accompany termination effects in setting the position model again overpredicts the polymerization rate. As of the maximum rate for styrene/DTAB microemulsions the monomer partitioning measurements (part 1) show, at low monomer loadings. this observation is consistent with our rough estimate Figures 9 and 10 show the kinetic profiles for nC4- (∼30 vol %) for the volume fraction of monomer in the MA and tC4MA microemulsions containing R)3wt% polymer particles at which glass transition sets in. Note of monomer initiated with V50 at 60 °C for two initiator that the slight difference in the location of the phase concentrations. For nC4MA, the maximum rate of po- boundary for styrene microemulsions made with H2O lymerization occurs at 29 ( 6% and 27 ( 3% conversion. and D2O(R)7.2 and 8.2 wt %, respectively) is not Similar to C6MA/DTAB microemulsions, the shift in the expected to have a significant impact on monomer position of the rate maximum from the theoretically partitioning. Furthermore, as discussed in section 3.1, predicted value of 39% down to an average of ∼28% can the nonlinear monomer partitioning at high styrene be attributed mainly to nonlinear monomer partitioning 3242 de Vries et al. Macromolecules, Vol. 34, No. 10, 2001

exhibited linear monomer partitioning (b ) 1). Appar- 28 ently, the higher propagation rate constant of nC4MA compared to styrene (1015 vs 342 M-1 s-1) is adequate to compensate for its high water solubility. Moreover, the value now quoted for τterm of styrene/DTAB microemulsions at R)3 wt % was obtained assuming that diffusion limitations to propagation are absent. Ultimately, however, the most interesting feature of Figures 9 and 10 is the manner in which the kinetic profiles of nC4MA and tC4MA track each other. As expected, due to their similar physical properties, the kinetic profile of tC4MA matches that of nC4MA up to ∼15% conversion after which it begins to fall below. This eventually leads to maximum rates of polymerization at 19 ( 5% and 20 ( 3% conversion for tC4MA. Following our previous discussion, we can only conclude that the drop in the rate of tC4MA polymerizations is due to diffusion limitations resulting from a glass transition. This result suggests that diffusion limita- Figure 9. Experimental and model calculated kinetics for nC4- tions may also be important for styrene microemulsion R) MA and tC4MA DTAB-based microemulsions containing polymerizations with low monomer content. 3 wt % monomer on a surfactant-free basis. The microemul- However, it is not very reassuring to note that our sions were initiated with V50: (b:nC4MA, 0.24 mM; 9:nC4- MA, 0.061 mM) and (O:tC4MA, 0.24 mM; 0:tC4MA, 0.061 monomer partitioning measurements (part 1) indicate mM). These initiator concentrations correspond to 0.25 and that up to 35 vol % of tC4MA is still present in the 0.625 wt % of the monomer. polymer particles at 15% conversion. As mentioned previously, ESR measurements24 for poly(methyl methacrylate) show that propagation becomes diffusion limited only at monomer volume fractions less than 30%. Nevertheless, there remains the remote possibility that nascent poly(tC4MA) or polystyrene particles be- come glassy at their earliest stages of growth. If this is true, monomer partitioning to the propagating particles may be diffusion-controlled. Unfortunately, the SANS monomer partitioning measurements offer no insight into this issue because it is not possible to isolate the growing particles that constitute only a small fraction of the total number of polymer particles.

5. Conclusions Application of our monomer partitioning results (part 1) to the polymerization kinetics of C6MA and nC4MA DTAB-based microemulsions highlights the effect of nonlinear monomer partitioning on the location of the maximum rate of polymerization. For microemulsion Figure 10. Experimental and model rate vs conversion compositions approaching the phase boundary, increas- profiles obtained by differentiating the data shown in Figure ingly nonlinear monomer partitioning shifts the maxi- 9 (A: nC4MA, 0.24 mM; C: nC4MA, 0.061 mM) and (B: tC4- mum rate from a predicted value of 39% conversion for MA, 0.24 mM; D: tC4MA, 0.061 mM). linear monomer partitioning to lower conversions of ∼30%. Nonlinear monomer partitioning cannot account (b ) 1.4). Interestingly, even though nC4MA has the for the low conversions (∼20%) at which the maximum same aqueous solubility as styrene, which is approxi- rate has been observed for styrene. A more careful, mately 10 times higher than that of C6MA (Table 1), albeit still approximate, analysis demonstrates that, in termination effects remain negligible. Indeed, the solid contradiction to the assumptions of our original kinetic curves in Figures 9 and 10 are the result of model model, biradical termination effects cannot be readily calculations obtained by conservatively assuming the dismissed. Although biradical termination is indeed same value for the characteristic time scale for termina- negligible for C6MA and nC4MA, and presumably for -6 tion (τterm ) (1.9 ( 0.6) × 10 s) as that for styrene/ tC4MA as well, it is the dominant mechanism that DTAB microemulsions at R)3 wt %. A literature value establishes the location of the maximum rate for sty- -1 -1 20 of ktr ) 0.02 M s , equal to that for styrene, was rene. The relative importance of termination is deter- also used for the chain-transfer constant of nC4MA mined to a large extent by the water solubility and monomer. Recall that these values for τterm and ktr fully reactivity of the monomer. In going from C6MA to accounted for the shift in the maximum rate to ∼20% styrene, the 10-fold increase in monomer water solubil- conversion for styrene/DTAB microemulsion with R) ity and 3-fold reduction in the propagation rate constant 3 wt %. Furthermore, note that unlike the present nC4- drastically increases the possibility of small free radicals MA microemulsions, for which nonlinear monomer encountering and terminating growing polymer par- partitioning alone would have shifted the maximum rate ticles. Up to this point, there is still no direct evidence to 32% conversion (Table 2), the styrene microemulsions for the role of glass transition during styrene micro- Macromolecules, Vol. 34, No. 10, 2001 Microemulsion Polymerization. 2 3243 emulsion polymerizations, and its effect is still uncer- In terms of these quantities, the steady-state results for tain. Nevertheless, the startling similarities and differ- / / N aq and N mic are ences in the kinetic profiles of nC4MA and tC4MA lead F + F us to believe that diffusion limitations may also play / 0 1 r1 0 1 N ) ≈ an important, yet so far unclear, mechanistic role during aq (mic) (1 + r )(1 + r ) - 1 (mic) r + r styrene microemulsion polymerizations. kads N 1 2 kads N 1 2 (A6) F F / 0 1 0 1 Appendix A N ) ≈ (A7) mic (mic) + + - (mic) + k (1 r1)(1 r2) 1 k r1 r2 Here we derive eq 16 for pterm/pprop using a coupled des des set of rate equations for the concentration of aqueous / Substituting these expressions in the rate equation for phase free radicals (N ), the concentration of grow- aq the concentration of growing particles, eq A3, one finds ing particles (N*), and the concentration of swollen / micelles with adsorbed oligomeric free radicals (N ). - aq ∂N* 1 r1/r2 The equations below are very similar to those for the )F (A8) ∂t 0 1 + r /r fate of transfer-generated radicals in emulsion polym- 1 2 erization, as discussed by Gilbert.26 To derive eq 16, it Hence we identify is sufficient to consider only initiator-derived oligomeric free radicals. Also, aqueous phase termination is not (part) (mic) pterm r1 kads kdes N* taken into account here. The extensions to also include ) ) (A9) (mic) (mic) transfer-generated radicals and aqueous phase termina- pprop r2 k k C N tion are straightforward but more involved and lead to ads p mon the same result for p /p . (mic) term prop This is identical to eq 16 upon setting k ) 1/τ . Oligomeric aqueous phase radicals are generated at des res F a rate 0 and adsorb on monomer-swollen micelles or Appendix B growing particles. Adsorption by dead particles is neglected. The second-order rate constants for adsorp- Explicit expressions for the monomer concentration (mic) in the micelles (C(mic)) and for the concentration of tion on micelles and particles are respectively kads mon (part) micelles (N) can be deduced from the monomer concen- and k . Adsorption on a growing particle leads to ads tration in the particles (eq 8) through mass balance instantaneous termination, but oligomeric radicals ad- equations. The composition of the initial microemulsion sorbed on micelles desorb back to the aqueous phase at is set by the initial volume fraction of monomer (φ ) a rate k(mic): mon,0 des and by the surfactant volume fraction (φsurf). Following / the analysis in part 1, the monomer and polymer volume ∂N aq / / / fractions during the polymerization reaction are ap- )F - k(mic)NN - k(part)N*N + k(mic)N ∂t 0 ads aq ads aq des mic proximated by (A1) φ ) (1 - f)φ (B1) where N is the concentration of monomer-swollen mi- mon mon,0 celles. Either micelles with adsorbed oligomeric free φ ) fφ (B2) radicals lose the radical through desorption, or the pol mon,0 radical propagates and becomes a growing particle: The monomer partitions between particles and micelles: / (mic) (part) ∂N mic (mic) / (mic) / (mic) / φ ) φ + φ (B3) ) k NN - k N - k C N mon mon mon ∂t ads aq des mic p mon mic (A2) Again following our previous paper on monomer partitioning, the extent of swelling of the micelles is char- Finally, the rate equation for the concentration of acterized by the parameter growing particles is R φ(mic) ∂N* ) (mic) / - (part) / ≡ mic ) + mon kpCmon N mic kads N aqN* (A3) x 1 (B4) ∂t Rmic,0 φsurf - The adsorption desorption process is fast on the time where Rmic,0 is the radius of the empty micelles, at zero scale of the polymerization reaction. Hence, we can use monomer content, and where we assume that the the steady-state approximation to calculate the concen- surfactant headgroup area does not depend on the trations of aqueous phase free radicals and micelles with extent of swelling of the micelles. The swelling of the adsorbed oligomeric free radicals from eqs A1 and A2. polymer particles is characterized by their monomer At this stage it is convenient to introduce the small volume fraction, dimensionless quantities r1 and r2, φ(part) (part) ) mon kads N* νmon (B5) r ) (A4) φ(part) + φ 1 (mic) N mon pol kads This is related to the monomer concentration in the (mic) (part) k C particles by C ) Cmonνmon, where Cmon is the con- r ) p mon (A5) mon 2 (mic) centration of pure monomer. Neglecting the small kdes amount of surfactant that is adsorbed on the particles, 3244 de Vries et al. Macromolecules, Vol. 34, No. 10, 2001 the two swelling parameters are related by the mass (11) Halnan, L. F.; Napper, D. H.; Gilbert, R. G. J. Chem. Soc., balance equation Faraday Trans. 1 1984, 80, 2851. (12) Yalkowsky, S. H.; Banerjee, S. Aqueous Solubility: Methods of Estimation for Organic Compounds; Marcel Dekker: New φ York, 1992. ) + mon,0 - f (13) Quian, R.; Wu, L.; Shen, D.; Napper, D. H.; Mann, R. A.; x 1 (1 - ) (B6) φsurf 1 νmon Sangster, D. F. Macromolecules 1993, 26, 2950. (14) McAskill, N. A.; Sangster, D. F. Aust. J. Chem. 1979, 32, 2611. (15) Maxwell, I. A.; Morrison, B. R.; Napper, D. H.; Gilbert, R. G. The above expressions directly relate the concentration Macromolecules 1991, 24, 1629. of monomer in the micelles, (16) Full, A. P.; Kaler, E. W.; Arellano, J.; Puig, J. E. Macromol- ecules 1996, 29, 2764. (17) Capek, I.; Juranicova, V. Eur. Polym. J. 1998, 34, 783. φ(mic) (18) Full, A. P.; Puig, J. E.; Gron, L. U.; Kaler, E. W.; Minter, J. C(mic) ) C mon ) C (1 - 1/x) (B7) R.; Mourey, T. H.; Texter, J. Macromolecules 1992, 25, 5157. mon mon (mic) + mon (19) Gilbert, R. G.; Smith, S. C. Theory of Unimolecular and φmon φsurf Recombination Reactions; Blackwell Scientific: Oxford, 1990. (20) Polymer Handbook, 3rd ed.; Brandrup, J., Immergut, E. H., Eds.; John Wiley & Sons: New York, 1989. and the concentration of monomer-swollen micelles (21) Yamada, B.; Kageoka, M.; Takayuki, O. Macromolecules 1991, 24, 5234. (22) Yamada, B.; Kageoka, M.; Otsu, T. Polym. Bull. 1992, 28, (mic) + φmon φsurf φsurf 1 75. N ) ) (B8) (23) O’Driscoll, K.; Amin Sanayei, R. Macromolecules 1991, 24, 4 4 2 πR3 πR3 x 4479. 3 mic 3 mic,0 (24) Yamada, B.; Westmoreland, D. G.; Kobatake, S.; Konosu, O. Prog. Polym. Sci. 1999, 24, 565. (25) Faldi, A.; Tirrell, M.; Lodge, T. P.; Vonmeerwall, E. Macro- to the monomer concentration in the particles, the molecules 1994, 27, 4184. composition variables φmon,0 and φsurf of the initial (26) Gilbert, R. G. Emulsion Polymerization: A Mechanistic Approach; Academic Press: San Diego, 1995. microemulsion, and to the extent of conversion. 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